The crystallographic symmetry test for the correctness of a set of phases.

One of the most important problems in the application of direct methods for large structures is to establish reliable consistency criteria for the correctness of a phasing trial. The introduction in the twin variables method [Bethanis, Tzamalis, Hountas, Mishnev & Tsoucaris (2000). Acta Cryst. A56, 105-111] of a new criterion based on the crystallographic symmetry consists of testing the phase extension and refinement algorithm by deliberately sacrificing the space-group-symmetry information in the auxiliary variable set then using its gradual re-appearance as a criterion for correctness. In the present paper, the crystallographic symmetry test has been used in the implementation of the twin algorithm in two different ways: (i) as an overall test throughout the iterations that is likely to reflect the correctness of the phasing procedure for each one of the extension trials in a macromolecular phasing environment; (ii) as a convenient criterion to determine the optimum cycle for freeing the initial phases used by the algorithm for the phase-extension procedure.

Ab initio determination of a crystal structure by means of the Schrödinger equation.

It is shown that the use of the Schrödinger equation may lead to the ab initio determination of the positions of the nuclei in a crystal, given a limited number of diffracted-beam intensities. In particular, it is shown that an extremely simplified Schrödinger equation in physical momentum space provides a sufficiently sound theoretical basis to develop an algorithm using diffraction data alone. This algorithm has been tested with a known 41 atom crystalline structure (not including hydrogen atoms) in space group P1. The extracted information is sufficient to determine the positions of all atoms. In addition, theoretical developments relevant to the connection between momentum space in quantum mechanics and diffraction theory have been formulated.

New developments of the TWIN algorithm for phase extension and refinement in disordered supramolecular structures.

A new development of the TWIN algorithm is described and used for phase extension/refinement in supramolecular complexes. A small number of phased reflections at low resolution are sufficient for the quasi-automated determination of all atomic coordinates, including disordered atoms.

Upgrading the twin variables algorithm for large structures.

Phase extension from lower to higher resolution by using an upgraded TWIN variables algorithm [Hountas & Tsoucaris (1995). Acta Cryst. A51, 754-763] in protein molecules with close to 1,000 non-H atoms is presented. Three points of this procedure are of particular interest. (i) The use of a set of auxiliary variables providing a satisfactory fit for many kinds of constraints: the new algorithm works efficiently despite the extreme 'dilution' of very limited initial phase information into a much larger set of auxiliary variables. (ii) The extension of this auxiliary variables set beyond the resolution of the observed data, which enhances the phase extension in a so-called 'super-resolution' sphere. (iii) The use of the crystallographic symmetry as a new figure of merit and as a reliable test for the correctness of the phase-extension process allows an efficient screening.